Shor-Term Load Forecaing Mehod: An Evaluaion Baed on European Daa J. W. Taylor and P. E. McSharry, Senior Member, IEEE IEEE Tranacion on Power Syem, 22, 223-229, 2008. Abrac-- Thi paper ue inraday elecriciy demand daa from 0 European counrie a he bai of an empirical comparion of univariae mehod for predicion up o a dayahead. A noable feaure of he ime erie i he preence of boh an inraweek and an inraday eaonal cycle. The forecaing mehod conidered in he udy include: ARIMA modeling; periodic AR modeling; an exenion for double eaonaliy of Hol-Winer exponenial moohing; a recenly propoed alernaive exponenial moohing formulaion; and a mehod baed on he principal componen analyi (PCA) of he daily demand profile. Our reul how a imilar ranking of mehod acro he 0 load erie. The reul were diappoining for he new alernaive exponenial moohing mehod and for he periodic AR model. The ARIMA and PCA mehod performed well, bu he mehod ha conienly performed he be wa he double eaonal Hol-Winer exponenial moohing mehod. Index Term Elecriciy demand forecaing, exponenial moohing, principal componen analyi, ARIMA, periodic AR. E I. INTRODUCTION LECTRICITY demand forecaing i of grea imporance for he managemen of power yem. Long-erm foreca of he peak elecriciy demand are needed for capaciy planning and mainenance cheduling []. Mediumerm demand foreca are required for power yem operaion and planning [2]. Shor-erm load foreca are required for he conrol and cheduling of power yem. Shor-erm foreca are alo required by ranmiion companie when a elf-dipaching marke i in operaion. There are everal uch marke in Europe and he US. For example, in Grea Briain, one hour-ahead foreca are a key inpu o he balancing marke, which operae on a rolling one hour-ahead bai o balance upply and demand afer he cloure of bi-laeral rading beween generaor and upplier [3, 4]. More generally, error in predicing elecriciy load ha ignifican co implicaion for companie operaing in compeiive power marke [5]. I i well recognized ha meeorological variable have a very ignifican influence on elecriciy demand (ee, for example, [6]). However, in online hor-erm forecaing J. W. Taylor i wih he Saïd Buince School, Univeriy of Oxford, Park End Sree, Oxford OX HP, UK (e-mail: jame.aylor@b.ox.ac.u. P. E. McSharry i wih he Deparmen of Engineering Science, Univeriy of Oxford, Park Road, Oxford OX 3PJ, UK (correponding auhor; el: 44-865-273095; fax: 44-865-273905; e-mail: parick@mcharry.ne). yem, mulivariae modeling i uually conidered impracical [7]. In uch yem, he lead ime conidered are le han a day-ahead, and univariae mehod can be ufficien becaue he meeorological variable end o change in a mooh fahion, which will be capured in he demand erie ielf. Indeed, univariae model are ofen ued for predicion up o abou four o ix hour ahead, and, due o he expene or unavailabiliy of weaher foreca, univariae mehod are omeime ued for longer lead ime. In a recen udy [8], mehod for hor-erm load forecaing are reviewed, and wo inraday load ime erie are ued o compare a variey of univariae mehod. One of he aim of hi paper i o validae he reul of ha udy. I concluded ha a double eaonal verion of Hol-Winer exponenial moohing wa he mo accurae mehod, wih a new approach baed on principal componen analyi (PCA) alo performing well. Uing ime erie of inraday elecriciy demand from 0 European counrie, we empirically compare he beer mehod idenified in [8] and alo he following wo new candidae mehod: an inraday cycle exponenial moohing mehod (ee [9]), and a new periodic AR approach, which we believe ha no previouly been conidered for elecriciy demand forecaing. All of he mehod are pecifically formulaed o deal wih he double eaonaliy ha ypically arie in demand daa. Thi eaonaliy involve inraday and inraweek eaonal cycle. Arificial neural nework (ANN) have feaured prominenly in he load forecaing lieraure (ee [0]). Their nonlinear and nonparameric feaure have been ueful for mulivariae modeling in erm of weaher variable. However, heir uefulne for univariae hor-erm load predicion i le obviou. Indeed, he reul in [8] for he ANN were no compeiive. Alhough we accep ha a differenly pecified neural nework may be ueful for univariae load modeling, in hi paper, for impliciy, we do no include an ANN. We would hope ha he beer performing mehod in our udy can erve a benchmark in fuure udie wih ANN. The paper i rucured a follow: Secion II decribe he elecriciy demand daa; Secion III decribe he mehod included in he udy; Secion IV preen he po-ample reul of a comparion of he hor-erm foreca wih lead ime up o one day ahead; and Secion V ummarie he reul and conclude he paper.
2 II. THE 0 EUROPEAN ELECTRICITY DEMAND SERIES Our daae conied of inraday elecriciy demand from 0 European counrie for he 30 week period from Sunday 3 April 2005 o Saurday 29 Ocober 2005. We obained halfhourly daa for ix of he counrie, and hourly daa for he remaining four. The fir 20 week of each erie were ued o eimae parameer and he remaining 0 week o evaluae po-ample accuracy of foreca up o 24 hour ahead. For he half-hourly erie, hi implie 6,720 obervaion for eimaion and 3,360 for evaluaion. For he hourly erie, 3,360 obervaion were ued for eimaion and,680 for evaluaion. We repreen he lengh of he inraday and inraweek cycle a and 2, repecively. For he half-hourly demand erie, = 48 and 2 = 336, and for he hourly erie = 24 and 2 = 68. Elecriciy demand on pecial day, uch a bank holiday, i very differen o normal day and can give rie o problem wih online forecaing yem. In pracice, ineracive faciliie end o be ued for pecial day, which allow operaor experience o upplemen or override he andard forecaing yem. In our udy, we moohed ou hee unuual obervaion by aking average of obervaion from he correponding period in he wo adjacen week. Half-Hourly TABLE I MEAN LOAD AND POPULATION FOR THE 0 COUNTRIES. Mean load (GW) Populaion (million) Mean load (W) per capia Belgium 9.4 0.4 906 Finland 8.3 5.2,588 France 47.8 60.9 786 Grea Briain 36.0 58.9 6 Ireland 2.9 4. 7 Porugal 5.2 0.6 492 Hourly Ialy 32.7 58. 563 Norway 2.0 4.6 2,599 Spain 25.5 40.4 630 Sweden 4.5 9.0,608 Table I li he 0 counrie along wih he mean elecriciy demand for he 30 week period and, in order o give a feel for he ize of he counrie, we alo li heir repecive populaion. For illuraive purpoe, in Fig., we plo he ime erie for France, Finland and Ireland. The decreae in French demand during Augu i due o he ummer vacaion. The Irih erie mainain a relaively conan level hroughou he 30 week period. By conra, he Finnih daa how a emporary level hif in he fir half of he erie. Thi wa due o inaciviy in he paper indury, which wa caued by a large conflic in conrac negoiaion beween he worker and employer. Demand (MW) 60000 40000 20000 0 France Finland Ireland 0 344 2688 4032 5376 6720 8064 9408 Half-hour Fig.. Half-hourly elecriciy demand in France, Finland and Ireland from Sunday, 3 April 2005 o Saurday, 29 Ocober 2005. Fig. 2 how he French erie for he fornigh in he middle of he 30 week period. Thi graph how a wihin-day eaonal cycle of duraion =48 period and a wihin-week eaonal cycle of duraion 2 =336 period. The weekday how imilar paern of demand, wherea Saurday and Sunday have differen level and profile. The inraweek and inraday feaure in Fig. 2 are ypical of hoe in all 0 erie. Demand (MW) 60000 40000 20000 0 0 48 96 44 92 240 288 336 384 432 480 528 576 624 672 Half-hour Fig. 2. Half-hourly elecriciy demand in France from Sunday, 0 June 2005 o Saurday, 23 June 2005. III. FORECASTING METHODS A. Simpliic Benchmark Mehod We implemened wo naïve benchmark mehod. The fir wa a eaonal verion of he random walk, which ake a a foreca he oberved value for he correponding period in he mo recen occurrence of he eaonal cycle. Wih wo eaonal cycle, i eem enible o focu on he longer cycle, o ha he predicion i conruced imply a he oberved value for he correponding period in he previou week. The foreca funcion i wrien a yˆ ( ), where y k = y + k i he 2 demand in period, and k i he foreca lead ime (k 2 ). The econd impliic benchmark ha we ued wa he imple average of he correponding obervaion in each of he previou four week. For hi mehod, he foreca funcion i yˆ = y + y + y y. ( ) 4 + k + k 2 + k 3 + 2 + k 4
3 B. Seaonal ARMA Modeling Double eaonal ARMA model are ofen ued a benchmark in load forecaing udie (e.g. [], [8], [], [2]). For each of he 0 load erie, we followed he Box- Jenkin mehodology o idenify he mo uiable model baed on he eimaion ample of 20 week. We conidered differencing, bu he reulan model had weaker diagnoic han model fied wih no differencing. The muliplicaive double eaonal ARMA model (ee [3], p. 333) can be wrien a φ p ( L) Φ P ( L ) Ω P ( L )( y c) = θ q ( L) Θ Q ( L ) ΨQ ( L ) ε 2 2 where c i a conan erm; L i he lag operaor; ε i a whie noie error erm; φ p, Φ,, θ P Ω P q, Θ and 2 Q Ψ are Q2 polynomial funcion of order p, P, P 2, q, Q and Q 2, repecively. The model can be expreed a ARMA ( p, q) ( P, ) (, ). Q P 2 Q2 We eimaed he model uing maximum likelihood wih he likelihood funcion baed on he andard Gauian aumpion. We conidered lag polynomial up o order hree. Thi choice wa made arbirarily, bu i i conien wih oher load forecaing udie, and i wa uppored by experimenaion wih everal of he erie. We baed model elecion on he Schwarz Bayeian Crierion, wih he requiremen ha all parameer were ignifican (a he 5% level). C. Periodic AR Model In inraday elecriciy demand ime erie, he inraday eaonal cycle i uually reaonably imilar for he five weekday, bu quie differen for he weekend. Thi implie ha he auocorrelaion a a lag of one day i ime-varying acro he day of he week. Such ime-variaion canno be capured in he eaonal ARMA model decribed in he previou ecion. A cla of model ha can capure hi feaure i periodic ARMA model. In hee model, he parameer are allowed o change wih he eaon [4]. Such model have been hown o be ueful for modelling economic daa (e.g. [5]). In he elecriciy conex, periodic model have been conidered wih ome ucce in udie inveigaing mehod for forecaing inraday ne imbalance volume [4] and daily elecriciy po price [6]. To ae he poenial for periodic ARMA model, we examined wheher he auocorrelaion a a pecified lag exhibied variaion acro he period of he day or he week. For example, for he half-hourly French erie, Fig. 3 how how he auocorrelaion a lag =48 varie acro he 2 =336 half-hour of he week. The fir period of he x-axi correpond o he fir period on a Sunday. The auocorrelaion value were calculaed from ju he 20-week in-ample period. Alhough he ample ize i no ufficienly large o conclude wih confidence, he variaion in he auocorrelaion in hi plo, and in imilar graph for he oher erie, uggeed o u ha here wa ome appeal in eimaing periodic ARMA model for our daa. Sudie have hown ha periodic MA erm are unneceary (ee [4], p. 28), and o for impliciy, in our work, we conidered only periodic AR model. More pecifically, we eimaed model wih periodiciy in he coefficien of AR erm of lag. The formulaion for hi mehod i preened in he following expreion: ( φ L)( φ ( ) L )( φ L )( y c) = ε 2 where d ( ) d ( ) 4 λ ( ) + ( ) ( ) ( ) i in 2iπ v = + i co 2iπ φ ( ) ω = w( ) w( ) i + κ i in 2iπ + υ co 2 2 i iπ d() and w() are repeaing ep funcion ha number he period wihin each day and week, repecively. For example, for he half-hourly erie, d() coun from o 48 wihin each day, and w() coun from o 336 wihin each week. ω, λ i, ν i, κ i, υ i, φ and φ are conan parameer. The periodic 2 φ, ue a imilar flexible fa Fourier form o parameer, ( ) ha employed in an analyi of he volailiy in inraday financial reurn [7]. For impliciy, we arbirarily choe o um from i= o 4 for all 0 erie. The parameer were eimaed uing maximum likelihood. 0.95 0.9 0.85 0.8 0 48 96 44 92 240 288 336 Period of he week (half-hour) Fig. 3. For he in-ample period of he France erie, lag 48 auocorrelaion eimaed eparaely for each period of he week. D. Double Seaonal Hol-Winer Exponenial Smoohing Exponenial moohing ha found widepread ue in auomaed applicaion, uch a invenory conrol. The eaonal Hol-Winer mehod ha been adaped in order o accommodae he wo eaonal cycle in elecriciy demand erie [8]. Thi involve he inroducion of an addiional eaonal index and an exra moohing equaion for he new eaonal index. The muliplicaive formulaion for he double eaonal Hol-Winer mehod i given in he following expreion: l = α ( y ( d w )) ( ) + α l () 2 d = δ ( y ( l w )) + ( δ ) d (2) 2 w = ω ( y ( l d )) + ( ω) w (3) k yˆ ( = l d ( ( ) + k w + k + φ y l d w (4) where l i he moohed level; d and w are he eaonal indice for he inraday and inraweek eaonal cycle, repecively; α, δ and ω are he moohing parameer; and
4 yˆ i he k ep-ahead foreca made from foreca origin (where k ). The erm involving he parameer φ, in he foreca funcion (4), i a imple adjumen for fir-order auocorrelaion. A rend erm wa included in he original formulaion, bu we found i no o be of ue for our 0 erie. An imporan poin o noe regarding he double eaonal Hol-Winer exponenial moohing approach i ha, by conra wih ARIMA modeling and he majoriy of oher approache o hor-erm demand forecaing, here i no model pecificaion required. Thi give he mehod rong appeal in erm of impliciy and robune. The iniial moohed value for he level and eaonal componen are eimaed by averaging he early obervaion. The parameer are eimaed in a ingle procedure by minimizing he um of quared one ep-ahead in-ample error. We conrained he parameer o lie beween zero and one. The reulan parameer for he 0 load erie are preened in Table II. For many of he erie, he value of φ i very high and he value of α i very low indicaing ha he adjumen for fir-order auocorrelaion ha, o a large degree, made redundan he moohing equaion for he level. I i alo inereing o noe ha, for a given erie, he value are imilar for he wo moohing parameer, δ and ω, for he eaonal indice. We alo implemened a verion of he mehod wih he opimized value of δ and ω conrained o be idenical, and wih α=0 o ha he level wa e a a conan value equal o he mean of he in-ample obervaion. Thi formulaion delivered predicion only marginally poorer han he full mehod given in expreion () o (4). Thi i omewha urpriing, given ha hi reformulaion of he mehod involve ju wo parameer. TABLE II FOR EACH OF THE 0 LOAD SERIES, PARAMETERS OF THE HOLT-WINTERS METHOD FOR DOUBLE SEASONALITY. Half-Hourly α δ ω φ Belgium 0.043 0.46 0.75 0.820 Finland 0.000 0.083 0.53 0.996 France 0.004 0.249 0.23 0.987 Grea Briain 0.002 0.36 0.68 0.970 Ireland 0.009 0.227 0.53 0.90 Porugal 0.094 0.20 0.20 0.77 Hourly Ialy 0.039 0.27 0.28 0.944 Norway 0.039 0.26 0.5 0.863 Spain 0.036 0.93 0.27 0.87 Sweden 0.022 0.223 0.34 0.928 From a heoreical perpecive, exponenial moohing mehod can be conidered o have a ound bai a hey have been hown o be equivalen o a cla of ae pace model [9]. The double eaonal Hol-Winer formulaion of expreion () o (4) can be expreed a a ingle ource of error ae-pace model. Thi model can be ued a he bai for producing predicion inerval. The moivaion ha led u o conider periodic AR model promped u o alo conider periodiciy in he parameer of he double eaonal Hol- Winer mehod. Diappoiningly, hi did no lead o improved accuracy, and o for impliciy we do no repor hee furher reul in Secion IV. The iue of periodiciy i addreed in he nex ecion in an alernaive exponenial moohing formulaion ha ha recenly been propoed. E. Inraday Cycle Exponenial Smoohing Model for Double Seaonaliy A feaure of he double eaonal Hol-Winer mehod i ha i aume he ame inraday cycle for all day of he week, and ha updae o he moohed inraday cycle are made a he ame rae for each day of he week. An alernaive form of exponenial moohing for double eaonaliy i preened by Gould e al. in [9]. I allow he inraday cycle for he differen day o be repreened by differen eaonal componen. In addiion, i allow he differen eaonal componen o be updaed a differen rae by uing differen moohing parameer. Our implemenaion of he mehod of Gould e al. involve he eaonaliy being viewed a coniing of he ame inraday cycle for he five weekday and a diinc inraday cycle for Saurday and anoher for Sunday. The day of he week are hu divided ino hree ype: weekday, Saurday and Sunday. By conra wih double eaonal Hol-Winer, here i no repreenaion in he formulaion for he inraweek eaonal cycle. Due o i focu on inraday cycle, we erm hi mehod inraday cycle exponenial moohing. For any period, he lae eimaed value of he hree diinc inraday cycle are given a c, c 2 and c 3, repecively. The formulaion require hree correponding dummy variable, x, x 2 and x 3, defined a follow: if ime period occur in a day of ype j x j = 0 oherwie Gould e al. preen heir approach in he form of a ae pace model, and we follow hi convenion in our preenaion of he model in expreion (5) o (7): 3 y = l + xi ci + ε (5), i= l = l + αε (6) 3 c i = ci, + ( =, 2, 3 γ ijx j ε i ) (7) j= where l i he moohed level; ε i an error erm; and α and he γ ij are he moohing parameer. (Expreion (6) and (7) can eaily be rewrien a recurive expreion, which i he more widely ued form for exponenial moohing mehod.) A wih he double eaonal Hol-Winer mehod, we eimaed he iniial moohed value for he level and eaonal componen by averaging he early obervaion. The parameer were eimaed in a ingle procedure by minimizing he um of quared one ep-ahead in-ample
5 error. All parameer were conrained o lie beween zero and one. The γ ij can be viewed a a 3 3 marix of parameer ha enable he hree ype of inraday cycle o be updaed a differen rae. I alo enable inraday cycle of ype i o be updaed even when he curren period i no in a day of ype i. Several rericion have been propoed for he marix of γ ij parameer (ee [9]). We included in our empirical udy wo form of he mehod; one involved eimaion of he marix of γ ij parameer wih he only rericion being ha he parameer lie beween zero and one, and he oher involved he addiional rericion of common diagonal elemen and common off-diagonal elemen. Gould e al. noe ha hee addiional rericion lead o he mehod being idenical o he double eaonal Hol-Winer mehod of expreion () o (4), provided even diinc inraday cycle are ued, inead of hree, a in our udy. In our dicuion of he po-ample forecaing reul in Secion IV, we refer o hi econd form of he model a he rericed form. We found ha he reul were ubanially improved wih he incluion of he adjumen for fir-order auocorrelaion ha wa ued in expreion (4) of he double eaonal Hol- Winer mehod. In Secion IV, we repor only he reul for hi improved form of he inraday cycle mehod. F. A PCA-Baed Mehod PCA provide a mean of reducing he dimenion of a mulivariae daa e o a maller e of orhogonal variable. Thee new variable are linear combinaion of he original variable. They are uncorrelaed and explain mo of he variaion in he daa, and, for hi reaon, hey are commonly refereed o a principal componen. A mehod baed on PCA ha recenly been propoed for hor-erm load forecaing [8]. The mehod aim o capure he inraday variaion in elecriciy demand, and i can be viewed a a developmen of he approach where a eparae model i buil for each of he period of he day (ee, for example, [20]). The mehod exploi he imilariy beween inraday obervaion in order o reduce he number of model o be conidered. Noe ha hi approach could eaily be exended o he mulivariae cae, if weaher relaed variable were available. In hi ecion, we preen only an overview of he mehod, a deail are provided in [8]. The mehod proceed by arranging he obervaion a an (n d ) marix, Y, where n d i he number of day in he eimaion ample. Each column conain obervaion for a paricular inraday period. PCA i applied o he column of Y o deliver componen ha are column of a new (n d ) marix. For each componen, a regreion model i buil uing day of he week dummie and quadraic rend erm. The model are hen ued o deliver a day-ahead foreca for each componen. Load predicion are creaed by projecing foreca of he componen back ono he Y pace. The mehod i refined, and peeded up, by focuing aenion on ju he principal componen. Crovalidaion i ued o opimie wo parameer: he number of principal componen and he lengh of he raining period ued in he PCA. In our udy, he cro-validaion employed he fir half of he 20 week in-ample period for eimaion and he econd half for evaluaion. We e, a opimal parameer, hoe delivering he minimum um of quared one ep-ahead foreca error for he raining daa. The error reuling from hi mehod exhibi erial correlaion. A for he Hol-Winer exponenial moohing mehod, he mehod benefi by he addiion of an AR model of he error proce. I i worh noing ha in [20] an error model wa alo employed in order o correc for erial correlaion reuling from he ue of eparae model for each hour of he day. Le E ( be he predicion error aociaed wih a k ep-ahead foreca made from origin. The errorcorrecion model i of he following form: E = α 0( + α( E + α2( E () where he α l ( are parameer eimaed eparaely for each lead ime, k, uing LS regreion applied o he eimaion ample. Finally, wih hi model pecificaion ha now include he error correcion erm above, cro-validaion i ued o opimie he number of principal componen and he lengh of he raining period ued in he PCA. For each of our 0 load erie, he opimal value are preened in Table III. TABLE III FOR THE PCA METHOD, THE OPTIMAL NUMBER OF TRAINING WEEKS AND NUMBER OF PRINCIPAL COMPONENTS FOR THE 0 LOAD SERIES. Half-Hourly Number of raining week Number of principal componen Belgium 7 2 Finland 9 9 France 4 9 Grea Briain 8 2 Ireland 8 0 Porugal 8 9 Hourly Ialy 3 5 Norway 8 6 Spain 0 7 Sweden 8 6 IV. POST-SAMPLE FORECASTING RESULTS We evaluaed po-ample forecaing performance from he variou mehod uing he mean abolue percenage error (MAPE) and he mean abolue error (MAE). Having calculaed he MAPE for each mehod a each foreca horizon, we hen ummarized each mehod performance by averaging he MAPE acro he 0 load erie. For he halfhourly erie, MAPE value were available for 48 half-hour lead ime, while, for he hourly daa, foreca were obviouly only available for 24 hourly lead ime. In order o average acro all he erie, we focued only on he 24 hourly
6 lead ime. The reuling Mean MAPE value are preened in Fig. 4. Averaging MAE value acro he 0 erie did no eem enible becaue he value ended o be ubanially higher for he erie correponding o higher level of elecriciy demand. In view of hi, for each mehod, we ummarized he MAE performance acro he 0 erie by averaging, for each lead ime, he raio of he mehod MAE o he MAE of he eaonal random walk benchmark mehod. The relaive performance of he mehod according o hi meaure were very imilar o hoe for he Mean MAPE. In view of hi, we do no plo hi addiional meaure here. We alo evaluaed he mehod uing he roo mean quared percenage error and roo mean quared error, bu we alo do no repor hee reul becaue he relaive performance of he mehod for hee meaure were very imilar o hoe for he MAPE. Mean MAPE 3.0% 2.5% 2.0%.5%.0% 0.5% 0.0% 0 3 6 9 2 5 8 2 24 Foreca horizon (hour) Seaonal Random Walk Inraday Cycle Exp Sm - unrericed Periodic AR Inraday Cycle Exp Sm - rericed Seaonal ARMA PCA Double Seaonal Hol Winer Exp Sm Fig. 4. For he 0 load erie, mean MAPE ploed again lead ime. The fir poin o noe i ha Fig. 4 doe no how he reul for he econd impliic benchmark mehod ha involved he imple average of he correponding obervaion in each of he previou four week. The reul for hi mehod were poorer han hoe for he eaonal random walk, and o for impliciy we oped o omi he reul from he figure. Turning o he more ophiicaed mehod, he figure how he double eaonal Hol-Winer mehod performing he be, followed by he PCA mehod and hen eaonal ARMA. Of he wo verion of he inraday cycle exponenial moohing mehod, he rericed form appear o be beer, which i conien wih he reul in [9]. However, he reul for boh form of hi mehod are diappoining. Thi i alo he cae for he periodic AR mehod. Our view i ha here i rong poenial for he ue of ome form of period model, bu ha a longer ime erie may be needed o eimae he periodiciy in he parameer. Wih regard o Fig. 3, he ue of only 20 week of daa implie ha only 9 obervaion were available o eimae he inraweek cyclical paern in he auocorrelaion for a given lag. In a imilar way, 20 week of daa i perhap oo lile o provide adequae eimae of periodic model parameer. We hould alo commen ha he ranking of he mehod wa really quie able acro he 0 erie, and ha he double eaonal Hol-Winer mehod wa conienly he be regardle of he error meaure ued for evaluaion. For he Finland load erie, which, a hown in Fig., conain level hif in he eimaion period, he po-ample MAPE value were relaively high for all mehod. However, i i inereing o noe ha, for hi erie, he ranking of he mehod wa imilar o ha hown in Fig. 4. A we explained in Secion I, univariae mehod end o only be ued for predicing load up o lead ime of abou four o ix hour. In Fig. 5, we focu more cloely on he poample hree hour-ahead reul for he hree mehod ha performed he be in Fig. 4. Fig. 5 how he hree-hour ahead Mean MAPE reul ploed again ime of day. The large Mean MAPE value occur for all hree mehod around 8am, and hi i becaue, around hi ime of day, demand end o be changing more rapidly han a oher period of he day. The plo how he double eaonal Hol-Winer mehod dominaing a almo all period of he day. The reul for he oher wo mehod are much cloer, wih he eaonal ARMA mehod maching he PCA mehod excep for he period around 8am. Mean MAPE 3.0% 2.5% 2.0%.5%.0% 0.5% 0.0% 00:00 03:00 06:00 09:00 2:00 5:00 8:00 2:00 00:00 Time of Day Seaonal ARMA PCA Double Seaonal Hol Winer Exp Sm Fig. 5. For hree hour-ahead predicion, Mean MAPE for he 0 load erie ploed again ime of day. V. SUMMARY AND CONCLUDING COMMENTS In hi paper, we have ued 0 ime erie of inraday elecriciy demand o compare empirically a number of univariae hor-erm forecaing mehod. One of he aim of he paper ha been o validae he finding in [8] uing a ubanially larger daae. In addiion o he mehod ha performed well in ha udy, we have alo conidered he inraday cycle exponenial moohing mehod (ee [9]) and a
7 new periodic AR approach. Our reul confirm he finding in [8]. All he ophiicaed mehod ouperformed he wo naïve benchmark mehod, and he be performing mehod wa double eaonal Hol-Winer exponenial moohing, followed by he mehod baed on PCA, and hen eaonal ARMA. The reul for he new inraday exponenial moohing mehod and for he periodic AR model were a lile diappoining. We upec ha he performance of he periodic AR model may improve wih ue of a longer ime erie. Our reaoning i ha ju 20 week of daa may well be inufficien o capure he inraweek periodiciy in a parameer. The ame commen can alo be applied o he inraday cycle exponenial moohing mehod becaue he approach involve he eimaion of a relaively complex parameerizaion, which enable a form of periodic moohing parameer. The ucce of he double eaonal Hol-Winer exponenial moohing mehod i impreive, paricularly in view of he mehod impliciy. Ongoing work i aiming o gain inigh ino he mehod, wih paricular focu on he implicaion of including he auoregreive error correcion erm wihin he formulaion. In erm of adviing praciioner, he double eaonal Hol-Winer mehod would eem o be very aracive, a i i imple o underand and implemen, and i ha been hown o be accurae for hor-erm load predicion. Furhermore, he mehod i alo appealing becaue of he exience of an underlying aiical model, which enable he calculaion of predicion inerval. Finally, we hould acknowledge ha, if weaher predicion are available, weaher-baed load forecaing mehod may well be more accurae beyond abou four o ix hour ahead. However, for horer lead ime, he beer of he univariae mehod conidered in hi paper hould be compeiive. In addiion, he univariae mehod have rong appeal, in erm robune, for online load predicion. VI. ACKNOWLEDGEMENTS The auhor are graeful o a number of people and organizaion for upplying he daa. Thee include Mark O Mallley (Univeriy College Dublin, Ireland), Shani Majihia (Naional Grid, UK), Juan Toro (Tranmarke, Spain), Rui Peana (REN, Porugal), Maari Uuialo (Fingrid, Finland), Mikkel Sveen (Markedkraf), Samuele Grillo (Univeriy of Genova, Ialy), and naional ranmiion yem operaor, including Eirgrid in Ireland, RTE in France, Elia in Belgium and Terna in Ialy. We are alo graeful for he helpful commen provided by paricipan a he RTE-VT Workhop held in Pari in May 2006 and he Energy Forecaing Workhop held in Rio de Janeiro in January 2007. We alo hank Marcelo Medeiro for hi deailed commen. VII. REFERENCES [] P. E. McSharry, S. Bouwman, and G. Bloemhof, "Probabiliic foreca of he magniude and iming of peak elecriciy demand," IEEE Tranacion Power Syem, vol. 20, pp. 66-72, 2005. [2] E. Gonzalez-Romera, M.A. Jaramillo-Moran, D. Carmona-Fernandez, "Monhly Elecric Energy Demand Forecaing Baed on Trend Exracion," IEEE Tranacion on Power Syem, vol. 2, pp. 946-953, 2006. [3] M.P. 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Brace, "Shor-run foreca of elecriciy load and peak," Inernaional Journal of Forecaing, vol. 3, pp. 6-74, 997. VIII. BIOGRAPHIES Jame W. Taylor i a Reader in Deciion Science a he Saïd Buine School, Univeriy of Oxford. Hi reearch inere include exponenial moohing, predicion inerval, quanile regreion, volailiy forecaing, energy forecaing and weaher enemble predicion. Parick E. McSharry i a Royal Academy of Engineering/EPSRC Reearch Fellow a he Deparmen of Engineering Science, Univeriy of Oxford, and a Senior Member of IEEE. He i currenly uppored by a Marie Curie Reearch Fellowhip, funded by he European Union Sixh Framework. Hi reearch inere include ignal proceing, daa analyi, complex dynamical yem, mahemaical modelling and forecaing.